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Mirrors > Home > MPE Home > Th. List > funcestrcsetclem5 | Structured version Visualization version GIF version |
Description: Lemma 5 for funcestrcsetc 16789. (Contributed by AV, 23-Mar-2020.) |
Ref | Expression |
---|---|
funcestrcsetc.e | ⊢ 𝐸 = (ExtStrCat‘𝑈) |
funcestrcsetc.s | ⊢ 𝑆 = (SetCat‘𝑈) |
funcestrcsetc.b | ⊢ 𝐵 = (Base‘𝐸) |
funcestrcsetc.c | ⊢ 𝐶 = (Base‘𝑆) |
funcestrcsetc.u | ⊢ (𝜑 → 𝑈 ∈ WUni) |
funcestrcsetc.f | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) |
funcestrcsetc.g | ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))) |
funcestrcsetc.m | ⊢ 𝑀 = (Base‘𝑋) |
funcestrcsetc.n | ⊢ 𝑁 = (Base‘𝑌) |
Ref | Expression |
---|---|
funcestrcsetclem5 | ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑁 ↑𝑚 𝑀))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcestrcsetc.g | . . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))) | |
2 | 1 | adantr 481 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))) |
3 | fveq2 6191 | . . . . . 6 ⊢ (𝑦 = 𝑌 → (Base‘𝑦) = (Base‘𝑌)) | |
4 | fveq2 6191 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (Base‘𝑥) = (Base‘𝑋)) | |
5 | 3, 4 | oveqan12rd 6670 | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) = ((Base‘𝑌) ↑𝑚 (Base‘𝑋))) |
6 | funcestrcsetc.n | . . . . . 6 ⊢ 𝑁 = (Base‘𝑌) | |
7 | funcestrcsetc.m | . . . . . 6 ⊢ 𝑀 = (Base‘𝑋) | |
8 | 6, 7 | oveq12i 6662 | . . . . 5 ⊢ (𝑁 ↑𝑚 𝑀) = ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) |
9 | 5, 8 | syl6eqr 2674 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) = (𝑁 ↑𝑚 𝑀)) |
10 | 9 | reseq2d 5396 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = ( I ↾ (𝑁 ↑𝑚 𝑀))) |
11 | 10 | adantl 482 | . 2 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = ( I ↾ (𝑁 ↑𝑚 𝑀))) |
12 | simprl 794 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝐵) | |
13 | simprr 796 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ 𝐵) | |
14 | ovexd 6680 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑁 ↑𝑚 𝑀) ∈ V) | |
15 | 14 | resiexd 6480 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ( I ↾ (𝑁 ↑𝑚 𝑀)) ∈ V) |
16 | 2, 11, 12, 13, 15 | ovmpt2d 6788 | 1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑁 ↑𝑚 𝑀))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ↦ cmpt 4729 I cid 5023 ↾ cres 5116 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 ↑𝑚 cmap 7857 WUnicwun 9522 Basecbs 15857 SetCatcsetc 16725 ExtStrCatcestrc 16762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 |
This theorem is referenced by: funcestrcsetclem6 16785 funcestrcsetclem7 16786 funcestrcsetclem8 16787 funcestrcsetclem9 16788 |
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